Decay rate and radial symmetry of the exponential elliptic equation

Let n ⩾ 3 , α , β ∈ R , and let v be a solution of Δ v + α e v + β x ⋅ ∇ e v = 0 in R n , which satisfies the conditions lim R → ∞ 1 log R ∫ 1 R ρ 1 − n ( ∫ B ρ e v d x ) d ρ ∈ ( 0 , ∞ ) and | x | 2 e v ( x ) ⩽ A 1 in R n . We prove that v ( x ) log | x | → − 2 as | x | → ∞ and α > 2 β . As a consequence we prove that there exists a constant R 0 > 0 such that if the solution v ( x ) is radially symmetric for | x | < R 0 and satisfies some gradient bound, then v is radially symmetric about the origin in R n .